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However, there is a non-constructive proof that shows that linkedness is decidable in polynomial time. The proof relies on the following facts: The set of graphs for which the answer is "yes" is closed under taking minors. I. e., if a graph G can be embedded linklessly in 3-d space, then every minor of G can also be embedded linklessly.
A non-constructive proof might show a solution exists without specifying either an algorithm to obtain it or a specific bound. Even if the proof is constructive, showing an explicit bounding polynomial and algorithmic details, if the polynomial is not very low-order the algorithm might not be sufficiently efficient in practice.
This theorem can be proven by using both a constructive proof, and a non-constructive proof. The following 1953 proof by Dov Jarden has been widely used as an example of a non-constructive proof since at least 1970: [4] [5] CURIOSA 339. A Simple Proof That a Power of an Irrational Number to an Irrational Exponent May Be Rational.
Constructive versus non-constructive [ edit ] As is often the case with probabilistic arguments, this theorem is nonconstructive and gives no method of determining an explicit element of the probability space in which no event occurs.
The classical theorem proving the existence of a bijection in such circumstances, namely the Cantor–Bernstein–Schroeder theorem, is non-constructive. It has recently been shown that the Cantor–Bernstein–Schroeder theorem implies the law of the excluded middle, hence there can be no constructive proof of the theorem. [4]
(Constructive proofs of the specific example above are not hard to produce; for example = and = are both easily shown to be irrational, and =; a proof allowed by intuitionists). By non-constructive Davis means that "a proof that there actually are mathematic entities satisfying certain conditions would not have to provide a method to ...
The presentation of the non-constructive proof without mentioning Cantor's constructive proof appears in some books that were quite successful as measured by the length of time new editions or reprints appeared—for example: Oskar Perron's Irrationalzahlen (1921; 1960, 4th edition), Eric Temple Bell's Men of Mathematics (1937; still being ...
From the other direction, there has been considerable clarification of what constructive mathematics is—without the emergence of a 'master theory'. For example, according to Errett Bishop's definitions, the continuity of a function such as sin(x) should be proved as a constructive bound on the modulus of continuity, meaning that the existential content of the assertion of continuity is a ...